🧮 algebra

1. **Problem Statement:** Given a hyperbola centered at the origin with equation $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ where $$a^2=9$$ and $$b^2=4$$, find the key points includ

1. **State the problem:** We need to find the value of $k$ such that the quadratic equation $$x^2 - (k + 3)x + 2k = 0$$ has two equal roots. 2. **Recall the condition for equal roo

1. The problem is to solve the equation $$\frac{2x+3}{x-1} = 4.$$\n\n2. We use the property of equations that if $$\frac{A}{B} = C,$$ then $$A = B \times C,$$ provided $$B \neq 0.$

1. **State the problem:** We are given a system of inequalities and points to check, and then asked to graph inequalities and describe graph shapes. 2. **Check points for inequalit

1. **Problem:** Determine if the ordered pair (2, 3) is a solution to the inequality $x + y < 7$. 2. **Formula:** Substitute $x=2$ and $y=3$ into the inequality.

1. **State the problem:** Simplify the expression $x - \frac{1}{x} - \frac{1}{x} + \frac{1}{x} + \frac{1}{x}$ and solve for $x$ if possible. 2. **Simplify the expression:** Combine

1. The first question asks: What do the green dots on the parabola represent? 2. The graph is a red parabola opening upward with vertex below the x-axis and crossing the x-axis at

1. The problem is to solve the equation represented by the red circled math expression. 2. Since the user did not provide the explicit equation, I will assume a common example: sol

1. **State the problem:** Given the function

1. **Problem:** Calculate $-18 \div 3$. 2. **Formula:** Division of integers follows the rule: dividing a negative by a positive gives a negative result, dividing two negatives giv

1. **State the problem:** Rushdi's father is currently five times as old as Rushdi. In three years, the father's age will be four times Rushdi's age. We need to find Rushdi's curre

1. **State the problem:** We need to complete the table for the equation $$y = -x + 1$$ by finding the corresponding $$y$$ values for $$x = -2, 0, 1$$. 2. **Formula used:** The equ

1. **State the problem:** We are given the equation of a line $$y = -x + 1$$ and a table with some values of $$x$$ and corresponding $$y$$ values. We need to complete the table and

1. **State the problem:** We want to understand the graph of the function $h(x) = \frac{2}{x+3} + 5$ and compare it to the original function $f(x) = -\frac{1}{x}$.\n\n2. **Recall t

1. **State the problem:** Solve for $v$ in the equation $$\sqrt{v + 11} - 5 = 2.$$\n\n2. **Isolate the square root term:** Add 5 to both sides to get $$\sqrt{v + 11} = 7.$$\n\n3. *

1. **State the problem:** Solve for $x$ in the equation $$12 + \sqrt{x} + 7 = 4$$ where $x$ is a real number. 2. **Combine like terms:** Add the constants on the left side:

1. **State the problem:** Solve for $x$ in the equation $$\sqrt{x - 3} = 5$$ where $x$ is a real number. 2. **Recall the formula and rules:** The square root function $\sqrt{y}$ is

1. **State the problem:** Solve for $x$ in the equation $$\sqrt{x - 3} = 5$$ where $x$ is a real number. 2. **Recall the property of square roots:** For any real number $a \geq 0$,

1. The problem asks to write a function representing the arithmetic sequence 1, 10, 19, 28, ... and graph it. 2. The formula for the nth term of an arithmetic sequence is:

1. The problem asks to graph the function $f(x) = 2\lfloor x \rfloor$ where $\lfloor x \rfloor$ is the floor function, which returns the greatest integer less than or equal to $x$.

1. The problem asks if the function $$K(x) = \frac{5^x}{\sqrt{3} \cdot 6^x}$$ is exponential and to rewrite it in the form $$K(x) = ab^x$$ if it is. 2. Recall that an exponential f